I'm a biologist in the process of modeling a fairly simple biological system using a system of ODEs. To verify the simulations, I'm attempting to obtain an analytical steady-state solution that I can check the simulations against. My attempts so far haven't borne fruit, so I thought I'd toss the question out to mathematicians. This is my first post, so apologies if the question isn't right for this site.

This system models the change in the substrate Sn or the microbial population X in a perfectly-stirred vessel with microbes acting upon a substrate S1 to produce S3, S4 and S7 when the kinetics of the chemical reactions are thermodynamically reversible.

Sn,in is the input concentration of Sn. Km and vmax are constants that describe the "affinity" of the microbe to S1 and the maximum rate of the reaction respectively and Keq,n is the thermodynamic equilibrium constant for the reaction S1 -> A Sn + B S7. I need to solve this system for Sn where n=1,3,4,7.

2 Answers
2

As Andrey mentioned, you shouldn't expect an analytical solution since you're dealing with a system of algebraic equation in several variables. (Just to be clear, what we're envisioning here is the system of equations you get by setting all the left-hand-sides to be zero). In your case, I believe you have 5 variables (X, $S_i$) and 5 equations, whose denominators can be cleared to make everything polynomial.

Such systems can be solved numerically (once you specify numerical values for your constants). One tool that I've used before is PHCpack, though for your purposes maybe Mathematica or something similar will be just fine.

Perhaps an expert can describe how to calculate resultants of your system of polynomials which will give you information on when the nature of the roots of this system change as you change your parameters...

already have numerical solutions using Python and SciPy. Not being a mathematician however, figuring out analytical solutions by setting LHS=0 seemed like the best way to check if the system was programmed correctly. An expression like $S_3 = f(S_1, S_4, S_7, X)$ for each of the variables is what I'm looking for...
–
Chinmay KanchiApr 22 '10 at 19:14

Right, though what you probably mean is $S_3=f(K_m,K_{eq},\dots)$, etc. But those kinds of expressions don't exist in closed form in elementary functions for general polynomial systems, unless you have a very special set of equations. It is possible to write down perturbative series solutions which will be valid in certain regimes, though. If that would be interesting to you I might give it some thought.
–
j.c.Apr 22 '10 at 22:14

The values for the constants are actually known. What I'm after is a solution for each variable in terms of the other variables. It should then be possible to compute the intersection(s) of the solution-spaces to determine if my numerical solution arrives at a valid steady-state, given a particular set of constants.
–
Chinmay KanchiApr 23 '10 at 1:19